login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A063037 Numbers without 3 consecutive equal binary digits. 9
0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 25, 26, 27, 36, 37, 38, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 73, 74, 75, 76, 77, 82, 83, 84, 85, 86, 89, 90, 91, 100, 101, 102, 105, 106, 107, 108, 109, 146, 147, 148, 149, 150, 153, 154, 155, 164, 165, 166 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Complement of A136037; intersection of A003796 and A003726. - Reinhard Zumkeller, Dec 11 2007

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

It appears (but has not as yet been proved) that the terms of {a(n)} can be computed recursively as follows. Let {c(i)} be defined for i>=4 by c(i) = 2c(i-1) + 1, if n is a multiple of 3, else c(i) = 2c(i-1) - 1, with c(4) = 1. I.e., {c(i)}={1,1,3,5,9,19,37,73,147,...}, for i=4,5,6,... . Let a(1)=1, a(2)=2, a(3)=3. For n>3, choose k so that F(k)-2<n<=F(k+1)-2, where F(k) denotes the k-th Fibonacci number (A000045). Then a(n)=c(k)+2a(F(k)-2)-a(2F(k)-n-3). This has been verified for n up to 1100. [From John W. Layman, May 26 2009]

EXAMPLE

The binary representation of 9 (1001) has no 3 consecutive equal digits.

MAPLE

isA063037 := proc(n)

    local bdgs, rep, d, i ;

    if n = 0 then

        return true;

    end if;

    bdgs := convert(n, base, 2) ;

    rep := 1;

    d := op(1, bdgs) ;

    for i from 2 to nops(bdgs) do

        if op(i, bdgs) = op(i-1, bdgs) then

            rep := rep+1 ;

        else

            rep :=1 ;

        end if ;

        if rep > 2 then

            return false;

        end if;

    end do:

    return true ;

end proc:

for n from 0 to 50 do

    if isA063037(n) then

        printf("%d, ", n) ;

    end if;

end do: # R. J. Mathar, Dec 18 2013

PROG

(PARI) { n=0; for (m=0, 10^9, x=m; t=1; b=2; while (x>0, d=x-2*(x\2); x\=2; if (d==b, c++; if (c==3, t=x=0), b=d; c=1)); if (t, write("b063037.txt", n++, " ", m); if (n==1000, break)) ) } [From Harry J. Smith, Aug 16 2009]

CROSSREFS

Cf. A000975.

A000045 [From John W. Layman, May 26 2009]

Sequence in context: A032878 A032845 A023776 * A201992 A236562 A157189

Adjacent sequences:  A063034 A063035 A063036 * A063038 A063039 A063040

KEYWORD

easy,nonn

AUTHOR

Lior Manor, Jul 05 2001

EXTENSIONS

Missing "less than" sign supplied in the conjectured recurrence; thanks to Franklin T. Adams-Watters for pointing this out John W. Layman, Nov 09 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 25 01:30 EDT 2017. Contains 284036 sequences.